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Astron. Astrophys. 332, 786-794 (1998)

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2. The steady equilibrium state

We model a sharp interface by two semi-infinite homogeneous regions separated by a nonuniform layer. In the Cartesian coordinate system we use, the x -axis is along the direction of inhomogeneity, i.e. the equilibrium quantities depend on the x -coordinate only. The non-uniform layer extends from [FORMULA] to [FORMULA] (see Fig. 1).

[FIGURE] Fig. 1. A cartoon of the single interface in Cartesian coordinates.

The magnetic field [FORMULA] is assumed to be homogeneous throughout the whole space whereas the plasma flow, with velocity [FORMULA] parallel to the magnetic field, exists only for [FORMULA]. In particular :

[EQUATION]

Thus the flow speed is discontinuous at [FORMULA], while the other physical quantities such as density [FORMULA] and temperature [FORMULA] are assumed to behave smoothly through the nonuniform layer.

In our treatment we ignore gravity, so that the magnetohydrostatic equation is simply given by

[EQUATION]

This means that the thermal pressure [FORMULA] is uniform and the plasma [FORMULA] parameter constant

[EQUATION]

where [FORMULA] while [FORMULA] [FORMULA] and [FORMULA] [FORMULA] are the squares of the Alfvén and the sound speed respectively.

The Alfvén speed is assumed to have a linear profile through the nonuniform layer and to be constant elsewhere :

[EQUATION]

Since [FORMULA] is constant (2), the speed of sound is simply proportional to [FORMULA]:

[EQUATION]

We also introduce the cusp speed [FORMULA] which is defined as

[EQUATION]

Besides the plasma [FORMULA] parameter, the configuration is also characterized by the temperature ratio [FORMULA], which can be written in terms of the Alfvén speeds in the two homogeneous regions as

[EQUATION]

According to the definition of the Alfvén speed and from (3) and (4) the density profile [FORMULA] in the nonuniform layer takes the following form :

[EQUATION]

For the rest of the paper, length, speed, density and magnetic field strength are non-dimensionalized with respect to [FORMULA], [FORMULA], [FORMULA] and [FORMULA] respectively. [FORMULA] and [FORMULA] are the y - and z -component of the wave vector of the MHD surface mode on the sharp interface.

The unperturbed plasma is taken to be ideal because the dissipative effects can be neglected on the time scale of the MHD wave propagation.

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© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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